= Solution
The <significance level> of a test is a bound $\alpha$ on its probability of rejecting a true null; for a composite null the bound is uniform over its parameter values. Its power at an alternative parameter is its rejection probability under that parameter.
The <Neyman-Pearson lemma> concerns a simple null density $f_0$ and a simple alternative density $f_1$ for the whole sample. Among tests of size at most $\alpha$, a most powerful test rejects where $f_1>kf_0$, accepts where $f_1<kf_0$, and if necessary randomizes on equality so that its size is $\alpha$. Let its rejection probability function be $\phi_*$, with $0\leq\phi_*\leq1$, and let $\phi$ be any competing test of size at most $\alpha$.
Pointwise,
$$
(\phi_* -\phi)(f_1-kf_0)\geq0:
$$
on the rejection side both factors are nonnegative, and on the acceptance side both are nonpositive. Integrating gives
$$
\int\phi_*f_1-\int\phi f_1
\geq k\left(\int\phi_*f_0-\int\phi f_0\right)\geq0.
$$
Thus the likelihood-ratio test has at least as much power as any competitor. Even when the sample has a continuous density, the <likelihood ratio> can have an atom, so boundary randomization may still be necessary in the general statement. For the exponential test below the statistic has a continuous distribution and no randomization is needed.
Back to article page
